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February  2012, 32(2): 353-379. doi: 10.3934/dcds.2012.32.353

Lie's reduction method and differential Galois theory in the complex analytic context

David Blázquez-Sanz 1, and  Juan J. Morales-Ruiz 2,

1. 

Universidad Sergio Arboleda, Calle 74 no. 14-14, Bogotá, D.C., Colombia
2. 

Departamento de Matemática e Informática Aplicadas a la Ingeniería Civil, ETSI Caminos, Canales y Puertos, Universidad Politécnica de Madrid, Profesor Aranguren s/n (Ciudad Universitaria) - 28040 Madrid, Spain

Received  December 2010 Revised  April 2011 Published  September 2011

This paper is dedicated to the differential Galois theory in the complex analytic context for Lie-Vessiot systems. Those are the natural generalization of linear systems, and the more general class of differential equations adimitting superposition laws, as recently stated in [5]. A Lie-Vessiot system is automatically translated into a equation in a Lie group that we call automorphic system. Reciprocally an automorphic system induces a hierarchy of Lie-Vessiot systems. In this work we study the global analytic aspects of a classical method of reduction of differential equations, due to S. Lie. We propose an differential Galois theory for automorphic systems, and explore the relationship between integrability in terms of Galois theory and the Lie's reduction method. Finally we explore the algebra of Lie symmetries of a general automorphic system.
Keywords: differential equations in the complex domain., Differential Galois theory.
Mathematics Subject Classification: Primary: 34M05; Secondary: 12H05, 14L99, 34A2.
Citation: David Blázquez-Sanz, Juan J. Morales-Ruiz. Lie's reduction method and differential Galois theory in the complex analytic context. Discrete & Continuous Dynamical Systems, 2012, 32 (2) : 353-379. doi: 10.3934/dcds.2012.32.353
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show all references

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J. Phys. A, 27 (1995), 3463-3474.  Google Scholar

[2]

J. Phys. A, 30 (1997), 4639-4649. doi: 10.1088/0305-4470/30/13/015.  Google Scholar

[3]

J. Phys. A, 31 (1998), 6605-6614. doi: 10.1088/0305-4470/31/31/008.  Google Scholar

[4]

Contemp. Math., 509, Amer. Math. Soc., Providence, RI, 2010.  Google Scholar

[5]

J. Lie Theory, 20 (2010), 483-517. Google Scholar

[6]

Lectures at the R.G.I. in Park City, Utah, 1991. Google Scholar

[7]

Lecture Notes in Mathematics, 1226, Springer-Verlag, Berlin, 1986.  Google Scholar

[8]

Napoli Series on Physics and Astrophysics, Bibliopolis, Naples, 2000.  Google Scholar

[9]

Rep. Math. Phys., 60 (2007), 237-258.  Google Scholar

[10]

Acta Appl. Math., 66 (2001), 67-87. doi: 10.1023/A:1010743114995.  Google Scholar

[11]

SIGMA Symmetry Integrability Geom. Methods Appl., 4 (2008), Paper 031, 18 pp.  Google Scholar

[12]

Acta Appl. Math., 70 (2002), 43-69. doi: 10.1023/A:1013913930134.  Google Scholar

[13]

(French) [Singular foliations of codimension one, Galois groupoids and first integrals], Ann. Inst. Fourier (Grenoble), 56 (2006), 735-779.  Google Scholar

[14]

"Algebraic, analytic and geometric aspects of complex differential equations and their deformations. Painlevé hierarchies,'' 15-20, RIMS Kôkyûroku Bessatsu, B2, Res. Inst. Math. Sci. (RIMS), Kyoto, 2007.  Google Scholar

[15]

Comment. Math. Helv., 83 (2008), 471-519. doi: 10.4171/CMH/133.  Google Scholar

[16]

(French) [Lessons on the general theory of surfaces. I, II] Reprint of the second (1914) edition (I) and the second (1915) edition (II), Les Grands Classiques Gauthier-Villars, Cours de Géométrie de la Faculté des Sciences, Éditions Jacques Gabay, Sceaux, 1993.  Google Scholar

[17]

Graduate Texts in Mathematics, 129, Readings in Mathematics, Springer-Verlag, New York, 1991.  Google Scholar

[18]

Compt. Rend. Acad. Sci. Paris, T CXVI, (1893), 964-965. Google Scholar

[19]

J. Math. Phys., 40 (1999), 3104-3122. doi: 10.1063/1.532749.  Google Scholar

[20]

Graduate Texts in Mathematics, No. 21, Springer-Verlag, New York-Heidelberg, 1975.  Google Scholar

[21]

Actualités Sci. Ind., No. 1251, Hermann, Paris, 1957.  Google Scholar

[22]

Amer. J. Math., 75 (1953), 753-824. doi: 10.2307/2372550.  Google Scholar

[23]

Pure and Applied Mathematics, 54, Academic Press, New York-London, 1973.  Google Scholar

[24]

J. Dynam. Control Systems, 1 (1995), 91-123. doi: 10.1007/BF02254657.  Google Scholar

[25]

Math. Ann. Bd., 25 (1885), 71-151. doi: 10.1007/BF01446421.  Google Scholar

[26]

Lepziger Berichte, 1893. Google Scholar

[27]

(German) Bearbeitet und herausgegeben von Georg Scheffers, Nachdruck der Auflage des Jahres 1893, Chelsea Publishing Co., Bronx, N.Y., 1971.  Google Scholar

[28]

Compt. Rend. Acad. Sci. Paris, T CXVI, (1893), 1233-1235. Google Scholar

[29]

Reprinted by Chelsea books, N.Y., 1967. Google Scholar

[30]

J. Math. Pures Appl., 4 (1839), 423-456. Google Scholar

[31]

(French) [The Galois groupoid of a foliation], in "Essays on Geometry and Related Topics," Vol. 1, 2, 465-501, Monogr. Enseign. Math., 38, Enseignement Math., Geneva, 2001.  Google Scholar

[32]

Chinese Ann. Math. Ser. B, 23 (2002), 219-226. doi: 10.1142/S0252959902000213.  Google Scholar

[33]

(French) [Lie Pseudogroups and Differential Galois Theory], Prepublications I.H.E.S., 2010. Google Scholar

[34]

(French) [Intrinsic Differential Geometry], Collection Enseignement des Sciences, No. 14, Hermann, Paris, 1972.  Google Scholar

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Phys. Rev. A (3), 52 (1995), 936-940. doi: 10.1103/PhysRevA.52.936.  Google Scholar

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Methods Appl. Anal., 8 (2001), 33-95, 97-111.  Google Scholar

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Methods Appl. Anal., 8 (2001), 113-120.  Google Scholar

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[39]

Ann. Sci. cole Norm. Sup. (4), 40 (2007), 845-884.  Google Scholar

[40]

Nagoya Math. J., 113 (1989), 173-179.  Google Scholar

[41]

Nagoya Math. J., 113 (1989), 1-6.  Google Scholar

[42]

Proc. Japan Acad. Ser. A Math. Sci., 73 (1997), 82-85. doi: 10.3792/pjaa.73.82.  Google Scholar

[43]

The Mathematical Society of Japan, 1956.  Google Scholar

[44]

(French) [Differential Galois theory and resummation], in "Computer Algebra and Differential Equations,'' 117-214, Comput. Math. Appl., Academic Press, London, 1990.  Google Scholar

[45]

An. Acad. Brasil. Ci., 35 (1963), 487-489.  Google Scholar

[46]

Aportaciones Matemáticas: Textos [Mathematical Contributions: Texts], 16, Sociedad Matemática Mexicana, México, 2001.  Google Scholar

[47]

J.-P. Serre, Géométrie algébrique et géométrie analytique, (French), 6 (): 1955.   Google Scholar

[48]

Séminaire Claude Chevalley, 3 (1958), 1-37. Google Scholar

[49]

Translated from the Japanese by the author, Translations of Mathematical Monographs, 82, American Mathematical Society, Providence, RI, 1990.  Google Scholar

[50]

J. Math. Phys., 25 (1984), 3155-3165. doi: 10.1063/1.526085.  Google Scholar

[51]

IEEE Trans. Automat. Control, 30 (1985), 266-272. doi: 10.1109/TAC.1985.1103934.  Google Scholar

[52]

in "Algebraic Geometry and Commutative Algebra,'' Vol. II, 771-789, Kinokuniya, Tokyo, 1988.  Google Scholar

[53]

Nagoya Math. J., 144 (1996), 1-58.  Google Scholar

[54]

Nagoya Math. J., 144 (1996), 59-135.  Google Scholar

[55]

(French) [On the equivalence of general differential Galois theories], C. R. Math. Acad. Sci. Paris, 346 (2008), 1155-1158.  Google Scholar

[56]

Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 328, Springer-Verlag, Berlin, 2003.  Google Scholar

[57]

(French), Ann. Sci. École Norm. Sup. (3), 9 (1892), 197-280.  Google Scholar

[58]

Ann. Sci. École Norm. Sup. (3), 10 (1893), 53-64.  Google Scholar

[59]

Compt. Rend. Acad. Sci. Paris, T. CXVI, (1893), 1112-1114. Google Scholar

[60]

Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys., 8 (1894), H1-H33.  Google Scholar

[61]

(French), Ann. Sci. École Norm. Sup. (3), 21 (1904), 9-85.  Google Scholar

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(French), Ann. École Norm. (3), 57 (1940), 1-60.  Google Scholar

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